Moire minibands in graphene heterostructures with almost commensurate √3 x √3 hexagonal crystals

We present a phenomenological theory of the low-energy moire minibands of Dirac electrons in graphene placed on an almost commensurate hexagonal underlay with a unit cell approximately three times larger than that of graphene. A slight incommensurability results in a periodically modulated intervalley scattering for electrons in graphene. In contrast to the perfectly commensurate Kekule distortion of graphene, such superlattice perturbation leaves the zero-energy Dirac cones intact, but is able to open a band gap at the edge of the first moire subbands, asymmetrically in the conduction and valence bands.